First slide

Elastic Modulie

Question

A material has normal density $ρ$ and bulk modulus K. The increase in the density of the material when it is subjected to an external pressure P from all sides is

Moderate

A

$\frac{p}{ρK}$
B

$\frac{K}{ρP}$
C

$\frac{ρP}{K}$
D

$\frac{ρK}{P}$

Solution

$\frac{ρ^{'}}{ρ} = \frac{\frac{M}{V - ∆ V}}{\frac{M}{V}} = \frac{V}{V - ∆ V}$
or $\frac{ρ^{'}}{ρ} = V {(V - ∆ V)}^{- 1}$
or $\frac{ρ^{'}}{ρ} = 1 + \frac{∆ V}{V}$
(Using Binomial theorem)
or $\frac{∆ V}{V} = \frac{ρ^{'}}{ρ} - 1 or \frac{∆ V}{V} = \frac{ρ^{'} - ρ}{ρ}$
Again,
$K = \frac{∆ P}{\frac{∆ V}{V}} = \frac{ρ ∆ P}{ρ^{'} - ρ} or ρ^{'} - ρ = \frac{ρ ∆ R}{K}$
But, $∆ P = P (given)$
$∴ ρ^{'} - ρ = \frac{ρP}{K}$

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